CatDat

Implication Details

Assumptions: connected, multi-terminal object

Conclusions: terminal object

This is an equivalence.

Reason: Let $(T_i)_{i\in I}$ be a multi-terminal object in a connected category $\mathcal{C}$. By definition of multi-terminal objects, for each object $C$, there are a unique index $i_C\in I$ and a unique morphism $C \to T_{i_C}$. Since the index $i_C$ is invariant under connected components, $I$ must be a singleton. The converse is trivial.